Proof that 4×4 is sometimes 10

TLDR: Base-16. Read on for a longer explanation if you haven’t heard that term before.

Yes, you read the title right. Under some specific circumstances and contexts, 4 times 4 is indeed 10. And this isn’t just abstract philosophizing or a gimmick – this is something engineers are directly working with and utilizing every day, in the real world. To understand why, we’ll be taking a meandering look at the very foundations of math, and what numbers actually mean and represent.

Have you ever noticed how “ten”, as a number, is qualitatively different from all the numbers that come before it. “Zero”, “One”, “Two”, “Three”, “Four”, “Five”, “Six”, “Seven”, “Eight”, “Nine” – these numbers all have their own unique symbol: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. A single digit that represents them.

Whereas “Ten”, despite having its own unique name, has no unique symbol of its own (that most people know of). If I asked you to write down the number ten, you would most likely write it as “10” – a composite of the numbers “1” and “0”.

Which begs the question – what does the concept of “ten” really mean? Is it the number “10” – the smallest number that doesn’t have a unique symbol of its own?  Or is it the number that comes immediately after 9, and represents things like the number of fingers that most humans have?

We’ve been so implicitly indoctrinated by our world, that most people don’t even see a distinction between the two. “Of course ‘ten’ means ‘10’ which is also the number that comes immediately after nine, and is the number of fingers most of us have. These are all identical concepts, there’s no difference between them!” 

But consider this – there is no objective reason for why we need the digit “9” at all. The system of math we’re all most familiar with, and take implicitly as “the truth”, is known as the decimal system. Or more precisely, base-10. Under the base-10 system, there is a unique symbol representing all numbers up to and including the number “nine”. After that, if you have 9 sheep and buy 1 more, you roll over to 10 – a composite of “1” and “0”.

This is the system we are most familiar with, but that doesn’t make it any more “right” than a different system. Consider for example, if we decided to adopt base-five. Under this system, you would count from 0, 1, 2, 3, 4 … and then immediately roll over to 10. Under the base-5 system, there is no unique symbol representing the number five – hence, if you want to express 4+1 – the number of fingers you have on one hand – you will have to roll over to 10. If you were to count out all of your fingers, you would count it as “1, 2, 3, 4, 10, 11, 12, 13, 14, 20”. Ie, the number of fingers you possess would no longer be 10, it would be 20… in a base-five world.

Hence the earlier question – what does the concept of “ten” really mean? Is it “10” – the smallest number that doesn’t have a unique symbol? Or is it the number of fingers most humans have? If you’re living in a society that has decided to use a system like base-five, these are two completely contradictory things. To them, 10 is indeed the smallest number that doesn’t have a unique symbol of its own – but the total number of fingers they possess is obviously 20, not 10.

Putting this into practice

Up to now, this all seems delightfully quirky, but also pointless armchair philosophy. Why on earth would we want to use any system besides base-ten? “The good lord gave us ten fingers, so clearly he must want us to base-ten! Or at the very least, we are all so used to base-ten, why would we ever use anything different?”

To understand why, look to computers and the chips that power them. You may have heard that computers are all about ones and zeros – this is literally true. The very building block of a computer is a tiny switch that is either turned on or turned off. Ie, a 1 or a 0. Anything in between is too hard to reliably distinguish. What this practically means is that every single number (as well as letters, and literally everything else) can only be read and written by computers as a series of 1s and 0s.

Hmmm, a number system that only allows for 0s and 1s? That sure sounds a lot like… base-two. Ie, binary. And this is exactly why base-two is even more practically useful to computer engineers than the base-ten system. Leading to a common joke within the industry: “There are 10 types of people in the world. Those who understand binary, and those who don’t.”

Admittedly, it gets pretty cumbersome if you have to read/write/display every single number in binary. Imagine if instead of “9”, you have to write “1001” instead. And instead of “102”, you have to read and make sense of “1100110”. The binary system does seem pretty darn inconvenient in such situations.

Hence why there is a second convention commonly used by computer engineers. Grouping together binary numbers into groups of 4. Imagine if you see the number 1100100 in binary. You can group them together into groups of 4, which gives you 0110,0100. Now look at the number within each group. 0110 in binary translates to 6 in decimal. If you don’t believe me, just try counting sheep in binary:

• 0 (base-2) == 0 (base-10)
• 1 (base-2) == 1 (base-10)
• 10 (base-2) == 2 (base-10)
• 11 (base-2) == 3 (base-10)
• 100 (base-2) == 4 (base-10)
• 101 (base-2) == 5 (base-10)
• 110 (base-2) == 6 (base-10)

And similarly, 0100 in binary translates to 4 in decimal. Put these 2 numbers together, and you get “64”, which is certainly a lot easier to read and write than 1100100.

There’s one problem with this though. A group of 4 binary numbers has 16 unique possibilities, not 10. The binary number “1110” corresponds to “14” in decimal. But if you simply wrote down “14”, do you mean “1110” – a single group of 4 binary digits? Or do you actually mean “0001,0100” – two groups of 4 binary digits, the first representing “1”, and the second representing “4”? This leads to even more confusion. If only we had a number system that has more than ten unique symbols….

To Infinity and Beyond

Fortunately, we do! There is nothing intrinsically magical about the base-10 system. There’s no reason at all why we can’t have a base-11 system, which has one additional unique symbol compared to base-10. When counting your numbers in a base-11 system, you would say “0123456789a10.” Note the additional “a” – that is the number that comes immediately after 9 in the base-11 system. You can pick any sound/picture you like to represent this new symbol – “a” is just a semi-arbitrarily chosen symbol that is widely used. “9+1” would now be equal to “a”, and “9+2” would now be equal to 10.

And this is just the start. You can choose to use a base-12 system that introduces “b” as a new unique symbol that comes immediately after “a”, and two digits after “9”. You can choose to use a base-13 system that introduces “c” as a new unique symbol that comes immediately after “b”. Heck, you could choose to invent a billion unique symbols, and use a base-billion system. In such a system, you could represent the population of the USA in just a single symbol! Though chances are, most people would have long forgotten what that symbol means.

This may all seem extremely strange to us, but that is only because we have had decades of indoctrination in the base-ten system. There are countless languages around the world, often dramatically different from each other, but each equally “valid” – the same goes for the base-system as well. If through an evolutionary quirk humans happened to possess 8 fingers, we would most likely be using a base-8 system today.

(And for a real mindfuck, consider the following: In such a world, there wouldn’t even exist a commonly used symbol for “eight”. “Eight” would simply be written as “10”. Hence, humans in this alternative universe would also call their number system “base-10”, even though you and I mean something completely different when we talk about base-10.)

Bases in the Digital World

Anyway, back to our universe and computers. A group of 4 binary digits can span the range from 0 to 15. But if I were to simply write down “15”, you as the reader would have no way of knowing whether I’m describing a single set of 4 binary digits, or two sets of 4 binary digits representing 1 and 5 individually. This ambiguity leads to inconvenience, confusion, and even faulty electronics.

Which is why computer engineers often use base-16 instead of base-2 or base-10. In base-16, every combination of 4 binary digits would correspond to a single base-16 digit:

• 0000 (base-2) == 0 (base-16) == 0 (base-10)
• 0001 (base-2) == 1 (base-16) == 1 (base-10)
• 0010 (base-2) == 2 (base-16) == 2 (base-10)
• 0011 (base-2) == 3 (base-16) == 3 (base-10)
• 0100 (base-2) == 4 (base-16) == 4 (base-10)
• 0101 (base-2) == 5 (base-16) == 5 (base-10)
• 0110 (base-2) == 6 (base-16) == 6 (base-10)
• 0111 (base-2) == 7 (base-16) == 7 (base-10)
• 1000 (base-2) == 8 (base-16) == 8 (base-10)
• 1001 (base-2) == 9 (base-16) == 9 (base-10)
• 1010 (base-2) == a (base-16) == 10 (base-10)
• 1011 (base-2) == b (base-16) == 11 (base-10)
• 1100 (base-2) == c (base-16) == 12 (base-10)
• 1101 (base-2) == d (base-16) == 13 (base-10)
• 1110 (base-2) == e (base-16) == 14 (base-10)
• 1111 (base-2) == f (base-16) == 15 (base-10)

And there you have it, you can now represent any group of 4 binary numbers using a single unique symbol. You can represent 1110010000100111 as “e427”, and there would be no confusion at all as to what that represents. And just to make it even more explicit, you can add a “0x” prefix to it, giving you “0xe427” – everyone in the industry would now know immediately that 0xe427 represents a base-16 number.

Which brings us back to the very original title. Imagine if you’re designing a computer-chip that can multiply numbers. Your goal is to design a tiny digital machine that when asked to compute 4×4, figures out that the answer is 16 (in base-10). But we’re talking about a digital machine, so all its answers are in the form of 0s and 1s. In order to express the number “16” (in base-10), it would have to produce a base-two answer of “10000”. Group this into sets of 4 digits, and we have 1,0000. Which you would write in base-16 as 10. Ergo, 4×4=10. Any other answer, as far as your tiny digital machine is concerned, is wrong.